Find the indefinite integral. ∫x(1−3x2)4dx

Question

Find the indefinite integral.  ∫x(1−3x2)4dx

Michele Grimsley · Accepted Answer

Step 1  Given: ∫x(1−3x2)4dx  for evaluating this limit we substitute  1−3x2=t  now differentiating it both side with respect to t  0−3(2x)dxdt=1   (∵ddx(kx2)=k(2x))  −6xdxdt=1  xdx=−dt6  now replacing xdx with −dt6 and (1−3x2) with t in given integral  Step 2  so,  ∫x(1−3x2)4dx=−∫t4dt6  =−16∫t4dt  (∵∫xmdx=xm+1m+1+c)  =−16(t4+14+1)+c  =−16(t55)+c  =−t530+c  now replacing t with 1−3x2  ∫x(1−3x2)4dx=−(1−3x2)530+c  hence, given integral is equal to −(1−3x2)530+c.

Fachur · Accepted Answer

Step 1: Use Integration by Substitution.
Let

u = 1 - 3 x^{2}, d u = - 6 x d x, t h e n x d x = - \frac{1}{6} d u

Step 2: Using u and du above, rewrite

\int x {(1 - 3 x^{2})}^{4} d x

.

\int - \frac{u^{4}}{6} d u

Step 3: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

.

- \frac{1}{6} \int u^{4} d u

Step 4: Use Power Rule:

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

.

- \frac{u^{5}}{30}

Step 5: Substitute

u = 1 - 3 x^{2}

back into the original integral.

- \frac{{(1 - 3 x^{2})}^{5}}{30}

Step 6: Add constant.

- \frac{{(1 - 3 x^{2})}^{5}}{30} + C

Find the indefinite integral. \int x(1-3x^{2})^{4}dx

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